Robust adaptive neural network control for a class ... - Semantic Scholar
Neurocomputing 125 (2014) 72–80
Contents lists available at ScienceDirect
Neurocomputing journal homepage: www.elsevier.com/locate/neucom
Robust adaptive neural network control for a class of uncertain nonlinear systems with actuator amplitude and rate saturations Ruyi Yuan n, Xiangmin Tan, Guoliang Fan, Jianqiang Yi Institute of Automation, Chinese Academy of Sciences, Beijing, PR China
a r t i c l e i n f o
a b s t r a c t
Available online 1 March 2013
An adaptive controller which is designed with a priori consideration of actuator saturation effects and guarantees H1 tracking performance for a class of multiple-input–multiple-output (MIMO) uncertain nonlinear systems with extern disturbances and actuator saturations is presented in this paper. Adaptive radial basis function (RBF) neural networks are used in this controller to approximate the unknown nonlinearities. An auxiliary system is constructed to compensate the effects of actuator saturations. Furthermore, in order to deal with approximation errors for unknown nonlinearities and extern disturbances, a supervisory control is designed, which guarantees that the closed loop system achieves a prescribed disturbance attenuation level so that H1 tracking performance is achieved. Steady and transient tracking performance are analyzed and the tracking error is adjustable by explicit choice of design parameters. Computer simulations are presented to illustrate the efficiency of the proposed controller. & 2013 Elsevier B.V. All rights reserved.
Keywords: Actuator saturation RBF neural network Adaptive control Robust control
1. Introduction All physical actuators in control systems have amplitude and rate limitations. For example, the elevator of an aircraft can only provide a limited force or torques in a limited rate. Actuator amplitude limitation or rate limitation constitutes a fundamental limitation on many linear or nonlinear control design techniques and has attracted the attention of numerous researchers. The controllers that ignore actuator limitations may cause the closed loop system performance to degenerate or even make the closed system unstable, and decrease the lifetime of the actuators, or damage the actuators. Higher performance may be expected if a controller is designed with a priori considering of the actuator saturation effects. The design of stabilizing controllers with a priori consideration of the actuator saturation effects for nonlinear systems with unknown nonlinearities and external disturbances is a challenging problem. Zhou [1] proposed an adaptive backstepping scheme to design an adaptive controller for a class of uncertain nonlinear single-input–single-output (SISO) systems in the presence of input saturations. To deal with saturations, an auxiliary system with the same order as that of the plant was constructed to compensate the effect of saturation. Farrell et al. [2–5] presented an adaptive backstepping approach and an online
n Correspondence to: Room 719, 95 Zhongguancun East Road, 100190 Beijing, PR China. Tel.: þ 86 10 62550985 16. E-mail addresses: [email protected] (R. Yuan), [email protected] (X. Tan), [email protected] (G. Fan), [email protected] (J. Yi).
0925-2312/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.neucom.2012.09.036
approximation based adaptive backstepping approach for unknown nonlinear systems with known magnitude, rate, bandwidth constraints on intermediate states or actuators without disturbance. Those approaches also used auxiliary systems for generating a modified tracking error to guarantee stability during saturation. Command filtered adaptive backstepping approaches [6–9] were also proposed to deal with the constraints on the control surfaces and the control states. For single input uncertain nonlinear systems in the presence of input saturation and unknown external disturbance, robust adaptive backstepping control algorithms were also developed by introducing a well defined smooth function and using a Nussbaum function which was used to compensate for the nonlinear term arising from the input saturation [10]. Dynamic inversion [11,12] approach is a widely used nonlinear control technique. However, the effects of actuator saturations have not been addressed with nominal dynamic inversion algorithm, so certain modifications are required. Tandale [13] proposed an adaptive dynamic inversion controller for a class of nonlinear systems with control saturation constraints. Enomoto [14] investigated the dynamic inversion control for nonlinear systems with control saturation constraints by Lyapunov synthesis. For a class of uncertain nonlinear dynamical systems in Brunovsky form, Lavretsky [15] proposed a dynamic inversion based adaptive control framework to provide stable adaptation in the presence of input constraints. The proposed design methodology can protect the control law from actuator position saturation. For a class of nonlinear systems which, in the presence of saturation, were controlled by nonlinear dynamic inversion
R. Yuan et al. / Neurocomputing 125 (2014) 72–80
controllers, an anti-windup compensation scheme was also proposed [16–18]. Neural network technique [19,20] was also used to handle actuator saturations problem. Calise [21] introduced a neural network based method termed as Pseudo-Control Hedging (PCH) for addressing a wide class of plant input characteristics such as actuator position limits, actuator rate limits, time delay, and input quantization. Chen et al. [22] also introduced a radial basis function neural network based controller for uncertain MIMO nonlinear systems with input saturations. The control design for nonlinear systems with actuator saturations was also be investigated by optimal control [23], nearly optimal control [24], nonlinear model predictive control [25] and fault tolerant scheme [26], etc. However, there are still few results for the control of uncertain nonlinear systems by taking actuator saturations into account in the controller design and analysis. In [27], the authors proposed an adaptive controller for MIMO nonlinear systems with control input limitations by using an auxiliary system and extended tracking errors which were used in neural network parameter update laws to compensate the effects of control input limitations. In this paper, for the control of a class of MIMO uncertain nonlinear systems in the presence of disturbances and actuator saturations, dynamic inversion [11] based controller which can generate constrained control signal is designed. Adaptive RBF neural networks are used to approximate unknown nonlinearities. An auxiliary system is constructed to compensate the effects of actuator amplitude and rate saturations. This auxiliary system and compensation scheme are different from [27], so that the extended tracking error in [27] is no longer needed. A supervisory control is designed to attenuate the effects of approximation errors and external disturbance so as to guarantee a H1 tracking performance. The performance of the closed loop system is obtained through Lyapunov analysis. The bounds of tracking errors can be adjusted by tuning the design parameters. The proposed controller can generate control signals satisfying actuator amplitude and rate limitations, and guarantee a H1 tracking performance of the closed loop system. The rest of this paper is organized as follows. In Section 2, the problem statement is presented. In Section 3, the adaptive control scheme is discussed, and the closed loop system performance is analyzed. A numerical example is shown in Section 4. Section 5 concludes the paper. Throughout this paper, 9 9 indicates the absolute value, J J indicates the Euclidean vector norm, and J J2 indicates the L2 norm.
2. Problem formulation Consider the class of MIMO systems described by the following differential equations: x_ i1 ¼ xi2 ^ x_ i,ri 1 ¼ xiri x_ iri ¼ f i ðxÞ þ
m X
ð1Þ g ij ðxÞuj þ di
i ¼ 1, . . . ,m
yðnÞ ¼ FðxÞ þGðxÞu þ d
ð2Þ m
ðnÞ def
..., where y ¼ ½y1 , . . . ,ym T A R is the output vector; y ¼ Pm ri ri ðr i Þ m mÞ T yðr r ¼ n; y ¼ d y =dt ; x ¼ ½x , . . . ,x , ..., 11 1r i m AR , i i¼1 1 i xm1 , . . . ,xmrm T A Rn is the state vector available for measurement; u ¼ ½u1 , . . . ,um A Rm is the control vector with u_ i rvimax
3. Design of adaptive controllers To begin, define t1 , . . . , tm as follows:
ti ¼ yðridi Þ þ
ri X
lij eðj1Þ , i ¼ 1, . . . ,m i
j¼1
i ¼ 1, . . . ,m are the reference signals, where yid , ei ¼ yid yi ði ¼ 1, . . . ,mÞ are the tracking errors, li1 , . . . , liri are parameters which make sure that the roots of the equation sri þ liri sri 1 þ þ li2 s þ li1 ¼ 0 are all in the open left-half complex plane. If FðxÞ and GðxÞ are known and the constrains on control inputs are ignored, then based on dynamic inversion algorithm, the control law: uc0 ¼ G1 ðxÞðFðxÞ þ sÞ
ð4Þ
can be applied to (2) to achieve the following asymptotically stable tracking: 2 3 r1 X ðr1 Þ ðj1Þ þ l e e 1j 1 6 1 7 6 7 j¼1 6 7 6 7 ^ ð5Þ 6 7¼0 6 7 r m 6 ðrm Þ X 7 ðj1Þ 5 4 em þ lmj em j¼1
in the case of no external disturbances. Because FðxÞ and GðxÞ are unknown vector and matrix respectively, the above control law (4) cannot be implemented in practice. Besides, there is no guarantee that uc0 satisfies the actuator constraints (3). It is well known that neural networks [19,20] can be used as universal approximators to approximate any continuous functions at any arbitrary accuracy as long as the network is big enough. In this work, in order to treat this tracking control design problem, radial basis function (RBF) neural networks are used to approximate the unknown functions, that is, f i ðxÞ, i ¼ 1, . . . ,m, and g ij ðxÞ, i,j ¼ 1, . . . ,m are approximated as follows: f i ðxÞ f^ i ðx9Hf i Þ ¼ HTfi Uf i ðxÞ,
i ¼ 1, . . . ,m i, j ¼ 1, . . . ,m
ð6Þ ð7Þ
M
which also can be rewritten in the following compact form:
9ui 9 r uimax ,
where uimax and vimax denote the actuator amplitude and rate limits respectively. FðxÞ ¼ ½f 1 ðxÞ, . . . ,f m ðxÞ A Rm , GðxÞ ¼ ½g ij ðxÞmm A Rmm (½mm represents a m m matrix) are continuous unknown functions of the state x. di denotes the external disturbances which RT 2 is unknown but bounded and satisfies d¼ 0 di dt o 1. m ½d1 , . . . ,dm A R . The control objective is to force yi to follow a given bounded reference signal yid in the presence of actuator saturations and extern disturbances. For (2) to be controllable, we assume that sðGðxÞÞ a0 for x in certain controllability region Uc A Rn , where sðGðxÞÞ denotes the minimum singular value of the matrix GðxÞ.
g ij ðxÞ g^ ij ðx9Hgij Þ ¼ HTgij Ugij ðxÞ,
j¼1
yi ¼ xi1 ,
73
½y1ðr1 Þ ,
ð3Þ
where Hf i A RMf i , Hgij A R gij are weight vectors, and Uf i ðxÞ A M RMf i ðxÞ, Ug ij ðxÞ A R gij ðxÞ are radial basis vectors, M f i , M g ij are the corresponding dimensions of the basis vectors. Denote 2 3 2 3 g^ 11 ðxÞ g^ 1m ðxÞ f^ 1 ðxÞ 6 7 6 7 ^ ^ HF Þ ¼ 6 ^ 7, Gðx9 & ^ Fðx9 ð8Þ HG Þ ¼ 4 ^ 5 4 5 ^f ðxÞ ^ m1 ðxÞ g^ mm ðxÞ g m ^ ^ HF Þ is an estimation of FðxÞ, and Gðx9 Then Fðx9 HG Þ is an estimation of GðxÞ.
74
R. Yuan et al. / Neurocomputing 125 (2014) 72–80
Using the approximation (8), and considering the control inputs constraints and extern disturbances, we modify the control law (4) as follows: #
uc ¼ G^ ðx9HG ÞðF^ ðx9HF Þ þ s þ g þ ud Þ
This means that by choosing appropriate parameters oi , zi , not only the actuators amplitude and rate saturations but also the dynamics of the actuators, like dampings and frequencies, can be integrated into the controller.
ð9Þ Remark 1. We known that for the first-order system:
u ¼ satðuc Þ
ð10Þ
g ¼ Cnn_
ð11Þ
where satðuc Þ represents the amplitude and rate limitations on uc . s ¼ ðt1 , . . . , tm ÞT is the robust control term. ud A Rm , which will be determined later, is a supervisor control used to attenuate the extern disturbance d. G# ðxÞ represents the generalized inversion [28] of GðxÞ. n ¼ ðx1 , . . . , xm ÞT A Rm is the state of the following constructed auxiliary system (12) which uses the difference of uc before and after amplitude and rate limitations as input: ^ n_ ¼ Cn þ Gðx9 HG ÞDu
ð12Þ
C ¼ diagðc1 , . . . ,cm Þ A Rmm , ci ði ¼ 1, . . . ,mÞ are positive parameters, Du ¼ uuc . n is used to compensate the effect of actuator saturations. uc is obtained according to certainty equivalence principle [29] which is widely used in adaptive control schemes. However, there is also no guarantee that uc satisfies the constraints (3), hence amplitude and rate limitations satðuc Þ are imposed on uc to generate u which satisfies the constraints (3). The amplitude and rate limitations on uc , i.e., satðuc Þ, can always be implemented by assuming a first-order model or second-order model for the dynamics of each component of uc , for example, u_ i ¼ satR ðoi ðsatA ðuci Þui ÞÞ or u€ i ¼ satR ðo2i ðsatA ðuci Þui Þ2zi oi u_ i Þ, where ui , uci are the i-th elements of u, uc respectively, oi , zi are positive constants, satR ðÞ, satA ðÞ represent the rate and amplitude saturation functions respectively. The function satR ðxÞ is defined as follows: 8 if x Z R > : R if x r R and satA ðxÞ is defined similarly. Fig. 1 gives visual descriptions for the first-order and second-order models respectively. In the linear range of the function satA ðxÞ and satR ðxÞ, the transfer function for the first-order model is Ui oi ¼ U ci s þ oi
ð13Þ
x_ ¼ f ðxÞ þ u
ð15Þ
if f ðxÞ is known, then we can construct a new system _ ðvÞ u ¼ vf
ð16Þ
with v as input and u as output. Eqs. (15) and (16) yield a closedloop system: _ ðxÞ ¼ vf _ ðvÞ xf
For system (17), the output x is totaly the same as the input v if and only if xð0Þ ¼ vð0Þ. The transform function from v to x is 1. Actually, the zeros of (16) is the poles of (15), and (16) is the zero assignment based inverse system of (15). Similarly, for the ^ auxiliary system (12), if we view the component Gðx9 HG ÞDu as the control input, then the new system: ^ HG ÞDu ¼ Ck þ k_ Gðx9
ð18Þ
^ HG ÞDu as the output is the zero with k as the input and Gðx9 assignment based inverse system of (12), (18) and (12) yield a closed loop system:
n_ þ Cn ¼ k_ þCk
ð19Þ
The output n is totaly the same as the input k if and only if nð0Þ ¼ kð0Þ. So n_ þ Cn or k_ þ Ck can be used as a prediction of ^ HG ÞDu which is the effect of actuator saturations and can be Gðx9 used to compensate the effect of actuator saturations. From (9)–(12), we have #
u ¼ satðuc Þ ¼ G^ ðx9HG ÞðF^ ðx9HF Þ þ s þ ud Þ
Remark 2. From (12) we also have that 0 1 Z tX m ci t @ c n i g^ ij ðxðnÞÞDuj ðnÞe dnA xi ðtÞ ¼ e xð0Þ þ 0 j¼1
ð14Þ
ð20Þ
By ignoring the term ud , u which satisfies the constraints (3) has the same form as uc0 which is obtained by dynamic inversion algorithm [11], which makes the controller be a adaptive neural network based dynamic inversion controller and the perfect tracking be available.
and the transfer function for the second model is
o2i Ui ¼ 2 U ci s þ 2zi oi s þ o2i
ð17Þ
¼ eci t xð0Þ þ
Z
t
m X
0 j¼1
g^ ij ðxðnÞÞDuj ðnÞeci ðtnÞ dn
Fig. 1. Schematic for amplitude and rate limitations. (a) First-order model and (b) second-order model.
R. Yuan et al. / Neurocomputing 125 (2014) 72–80
r eci t 9xð0Þ9 þ
Z
t
m X
0 j¼1
r eci t 9xð0Þ9 þ
m ci
9g^ ij ðxðnÞÞ9Duj ðnÞ9Þeci ðtnÞ dn
sup 9g^ ij ðxðnÞÞ9 sup 9Duj ðnÞ9
0rnrt
ð21Þ
0rnrt
where Duj ðnÞ ¼ uj ðnÞucj ðnÞ. Because ci is positive, the auxiliary system is bounded input bounded output stable. Remark 3. Although the true value of GðxÞ are invertible accord^ ing to the assumption, the estimate matrix Gðx9 HG Þ may become singular during the adaptive process, so Moore–Penrose general^ ized matrix inverse [28] of Gðx9 HG Þ is used. Remark 4. In the case of known FðxÞ and GðxÞ and free of external disturbances, the control low
75
where udi , di ,wi are the i-th element of ud , d, and w respectively. ~ ¼ H Hn , H ~ g ¼ Hg Hn , then Defining ei ¼ ½ei , . . . ,eiðri 1Þ T , H fi fi fi g ij ij ij Eq. (31) can be rewritten in the following form: 0 1 m X T T ~ ~ e_ i ¼ Ai ei þ Bi @H f i Uf i ðxÞ þ ð32Þ ðH g ij Ug ij ðxÞÞuj udi di þ wi A j¼1
where 2
0 6 0 6 6 ^ Ai ¼ 6 6 6 0 4 li1
1
0
0
0
1
^
^
&
0
0
li2
li3
3
0 7 7 7 ^ 7, 7 1 7 5 liri
2 3 0 607 6 7 6 7 7 Bi ¼ 6 6^7 6 7 405 1
ð22Þ
For the i-th subsystem of (26), the following theorem can be obtained.
uc ¼ G1 ðxÞðFðxÞ þ s þ gÞ
ð23Þ
Theorem 1. For the i-th subsystem of (26), if we select the control law (10), and the following parameters update laws and udi
g ¼ Cnn_
ð24Þ
_ ¼ G U ðxÞBT P e H fi fi fi i i i
ð33Þ
n_ ¼ Cn þ GðxÞDu
ð25Þ
_ g ¼ Gg Ug ðxÞBT P e u H i i i j ij ij ij
ð34Þ
u ¼ satðuc Þ with
can be applied to System (2) to obtain the asymptotically stable tracking (5).
udi ¼
In the following, we will specify the update laws for the RBF parameters Hf i ði ¼ 1, . . . ,mÞ, Hg ij ði,j ¼ 1, . . . ,mÞ and the supervisor control ud , so that desired tracking performance can be achieved. Applying the control law (9) and (10) to System (2) yields
then the following H1 tracking performance can be obtained: Z T ~ ð0Þ ~ T ð0ÞG1 H eTi Q i ei dt r eTi ð0ÞPi ei ð0Þ þ H fi fi fi 0 Z m T X T ~ ð0ÞG1 H ~ g ð0Þ þ r2 H R2i dt ð36Þ þ g ij g ij i ij
^ ^ HF ÞFðxÞ þðGðx9 sYðrÞ ¼ Fðx9 HG ÞGðxÞÞuud d Define the optimal approximation weight f i ði ¼ 1, . . . ,mÞ, g ij ði,j ¼ 1, . . . ,mÞ as follows: " # Hn ¼ arg min sup 9f ðxÞf^ ðx9H Þ9 fi
i
Hf i A OF x A Uc
i
fi
" n
Hg ij ¼ arg min
Hgij A OG
for
ð27Þ
sup 9g ij ðxÞg^ ij ðx9Hgij Þ9
ð28Þ
x A Uc
According to universal approximation property of neural networks [19,20], the following assumption is reasonable: Assumption 1. The minimum approximation error is square integrable, i.e., Z T wT w dt o 1 ð30Þ 0
Using the optimal approximation for FðxÞ, GðxÞ, the i-th subsystem of (26) can be rewritten as ri X
lik eðk1Þ ¼ f^ i ðx9Hf i Þf^ i ðx9Hnf i Þudi di þ wi i
k¼1
þ
m X j¼1
ð35Þ
0
where Gf i , Ggij ðj ¼ 1, . . . ,mÞ are positive definite diagonal matrices,
ri ði ¼ 1, . . . ,mÞ are positive parameters representing for prescribed def disturbance attenuation levels, Ri ¼ di þ wi , Q i A Rmm is arbitrary
symmetric positive definite matrices, Pi A Rmm is the symmetric positive definite solution of the following Lyapunov equation: Pi Ai þ ATi Pi ¼ Q i
#
where OF , OG , Uc denote the sets of suitable bounds on Hf i , Hg ij , and x respectively. Hnf i ði ¼ 1, . . . ,mÞ, Hngij ði,j ¼ 1, . . . ,mÞ are constant vectors. The optimal approximations for FðxÞ and GðxÞ are ^ ^ Hn Þ, Gðx9 denoted as Fðx9 HnG Þ respectively. Define the minimum F approximation error as h i def ^ ^ w ¼ Fðx9 HnF ÞFðxÞ þ Gðx9 HnG ÞGðxÞ u ð29Þ
iÞ eðr i þ
j¼1
ð26Þ vectors
1 T e P i Bi 2r2i i
n
ðg^ ij ðx9Hgij Þg^ ij ðx9Hgij ÞÞuj
ð31Þ
ð37Þ
Proof. Define the Lyapunov function Vi for the i-th subsystem as follows: Vi ¼
m 1 T 1 ~ T 1 ~ 1X ~ T G1 H ~g ei Pi ei þ H H f i Gf i H f i þ ij 2 2 2 j ¼ 1 gij gij
ð38Þ
The time derivative of Vi is m X 1 T ~_ þ ~_ g ~ T G1 H ~ T G1 H H V_ i ¼ ðe_ i Pi ei þ eTi Pi e_ i Þ þ H fi fi fi g ij g ij ij 2 j¼1
¼
1 T T ~ T UF ðxÞBT P e ðe A P e þ eTi Pi Ai ei Þ þ H i i i fi 2 i i i i m m X X ~ T U ðxÞBT P e u þ H ~ T G1 H ~_ þ ~_ g ~ T G1 H þ H H G fi g ij fi fi g ij g ij i i i j ij i¼1
j¼1
1 þ ðudi þ Ri ÞðBTi Pi ei þeTi Pi Bi Þ 2 1 1 1 r eTi Qei þ Ri ðBTi Pi ei þ eTi Pi Bi Þ 2 eTi Pi Bi BTi Pi ei 2 2 2r 2 1 1 1 1 r eTi Qei þ r2i R2i ri Ri eTi Pi Bi 2 2 2 ri 1 T 1 2 2 r ei Qei þ ri Ri 2 2
ð39Þ
Integrating both sides of the above inequality from 0 to T yields Z Z r2 T 2 1 T T ei Q i ei dt rV i ð0ÞV i ðTÞ þ i R dt ð40Þ 2 0 2 0 i
76
R. Yuan et al. / Neurocomputing 125 (2014) 72–80
Since VðTÞ 4 0, the inequality (40) implies the following inequality: Z Z r2 T 2 1 T T ei Q i ei dt rV i ð0Þ þ i R dt ð41Þ 2 0 2 0 i According to the definition of Vi, the following inequality is obtained: Z T ~ ð0Þ ~ T ð0ÞG1 H eTi Q i ei dt r eTi ð0ÞPi ei ð0Þ þ H fi fi fi 0
þ
m X
T
~ ð0ÞG1 H ~ g ð0Þ þ r2 H g ij g ij i ij
j¼1
This is (36).
Z
T 0
R2i dt
ð42Þ
&
Finally, for the nonlinear system (2), the following theorem can be obtained.
definite solutions of the following Lyapunov equations: Pi Ai þ ATi Pi ¼ Q i ,
A scheme of the adaptive H Fig. 2.
ð43Þ
1
ð51Þ
tracking control scheme is shown in
P Proof. Define the Lyapunov function V ¼ m i ¼ 1 V i where Vi is defined in (38). According to the definition of Vi and Theorem 1, it is easy to obtain (50). & Corollary 1. The closed loop system is stable and the steady tracking errors satisfy limt-1 ei ¼ 0,i ¼ 1, . . . ,m, i.e., limt-1 9yid ðtÞyi ðtÞ9 ¼ 0. The bound of the transient tracking errors will be given by Jei J22 r
Theorem 2. For the nonlinear system (2), if we select the control law u ¼ satðuc Þ
i ¼ 1, . . . ,m
~ ð0ÞÞ ~ T ð0ÞG1 H 2ðeTi ð0ÞPi ei ð0Þ þ H fi fi fi
lmin ðQ i Þ 2
R 1 ~ 2 T 2 ~T j ¼ 1 H g ij ð0ÞGg ij H g ij ð0Þ þ ri 0 Ri dt
Pm
þ
lmin ðQ i Þ
ð52Þ
where lmin ðQ i Þ represents the minimum eigenvalue of matrix Q i .
with Proof. From (39), it can be obtained that
#
uc ¼ G^ ðx9HG ÞðF^ ðx9HF Þ þ s þ g þ ud Þ
ð44Þ
g ¼ Cnn_
ð45Þ
^ n_ ¼ Cn þ Gðx9 HG ÞDu
ð46Þ
_ ¼ G U ðxÞBT P e , i ¼ 1, . . . ,m H fi fi fi i i i
ð47Þ
e2i rJei J2 r
_ g ¼ Gg Ug ðxÞBT P e u , i,j ¼ 1, . . . ,m H i i i j ij ij ij
ð48Þ
hence
1 T e Pi Bi , udi ¼ 2r2i i
i ¼ 1, . . . ,m
ð49Þ
then the following H1 tracking performance can be obtained: Z T m X ~ T ð0ÞG1 H ~ ð0Þ eT Qe dt r eT ð0ÞPeð0Þ þ H fi fi fi 0
i¼1
þ
m X i,j ¼ 1
~ T ð0ÞG1 H ~ g ð0Þ þ H g ij g ij ij
m X i¼1
r2i
Z 0
T
R2i dt
ð50Þ
where Gf i ði ¼ 1, . . . ,mÞ, Gg ij ði,j ¼ 1, . . . ,mÞ are positive definite diagonal matrices, ri ði ¼ 1, . . . ,mÞ are positive parameters representing for prescribed disturbance attenuation levels, e ¼ ½eT1 , . . . ,eTm T , Ri def ¼ di þ wi , Q ¼ diagðQ 1 , . . . ,Q m Þ and Q i A Rmm ði ¼ 1, . . . ,mÞ are arbitrary symmetric positive definite matrices, P ¼ diag ðP1 , . . . ,Pm Þ and Pi ði ¼ 1, . . . ,mÞ are the symmetric positive
Δ
V_i r 12 lmin ðQ i ÞJei J2 þ 12r2i R2i
ð53Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi _ the trackV i is negative whenever Jei J Z ri 9Ri 9= lmin ðQ i Þ. Hence pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ing error will stay in the region fei j Jei J r ri 9Ri 9= lmin ðQ i Þg. Obviously 2V_i þ r2i R2i lmin ðQ i Þ
ð54Þ
RT 2ðV i ð0ÞV i ðTÞÞ þ r2i 0 R2i dt lmin ðQ i Þ 0 R 2 T 2 2V i ð0Þ þ ri 0 Ri dt r lmin ðQ i Þ 2 ~ ð0Þ ~ T ð0ÞG1 H ðeT ð0ÞPi ei ð0Þ þ H ¼ fi fi fi lmin ðQ i Þ i R P 1 ~ 1 2 T 2 ~T 2ð m j ¼ 1 H g ij ð0ÞGg ij H g ij ð0Þ þ 2 ri 0 Ri dtÞ ð55Þ þ lmin ðQ i Þ RT 2 RT 2 then Assumption 1 implies 0 wi dt o 1, 0 Ri dt ¼ RT 2 0 ðwi di Þ dt o 1. Eq. (55) means ei A L2 . According to Barbalat lemma, limt-1 ei ¼ 0. & Jei J22 ¼
Z
T
e2i dt r
Remark 5. According to Theorem 1, the i-th subsystem achieves a H1 tracking performance with a prescribed disturbance
Δ
Fig. 2. The overall structure of adaptive H1 tracking control scheme.
R. Yuan et al. / Neurocomputing 125 (2014) 72–80
77
attenuation level ri , i.e., the L2 gain from wi to the tracking error ei is equal or less than ri .
networks as Gf 1 ¼ Gf 2 ¼ ¼ Gg11 ¼ Gg12 ¼ Gg21 ¼ Gg 22 ¼ diagð1,1, 1,1,1,1,1,1,1,1,1Þ and set the parameters update laws as follows:
Remark 6. Because the parameters lij , j ¼ 1, . . . ,r i can make Ai be a Hurwitz stable matrix, there exists unique symmetric positive definite matrix Pi satisfying Lyapunov equation (37).
_ ¼ G U ðxÞBT P e _ ¼ G U ðxÞBT P e , H H f1 f1 f1 f2 f2 f2 1 1 1 2 2 2
Remark 7. The bound for Jyid ðtÞyi ðtÞJ2 is an explicit function of the design parameters. According to Corollary 1, this bound ~ ð0Þ, depends on the initial estimate errors H fi ~ g ð0Þ ðj ¼ 1, . . . ,mÞ. The closer the initial estimates H ð0Þ, H fi ij Hg ij ð0Þ ðj ¼ 1, . . . ,mÞ to the true values Hnf i ð0Þ, Hngij ð0Þ ðj ¼ 1, . . . ,mÞ, the better the transient performance. The effects of initial estimate errors on this bound can be decreased by increasing the values of the diagonal adaptation gain matrices Gf i , Ggij ðj ¼ 1, . . . ,mÞ and by choosing positive definite symmetric matrix Q i with larger minimum eigenvalue. On the other hand, this bound also depends on the external disturbances and neural network approximation errors. The effects of extern disturbances and neural network approximation errors on the transient performance can be reduced by decreasing ri and increasing lmin ðQ i Þ. Large ri represents low disturbance attenuation level while small ri represents high disturbance attenuation level.
_ g ¼ Gg Ug ðxÞBT P e u , H _ g ¼ Gg Ug ðxÞBT P e u H 2 2 2 1 2 2 2 2 21 21 21 22 22 22
Remark 8. If Du ¼ 0 or Du tends to zero as t tends to infinity, then xi -0. 4. Numerical example In this section, we illustrate the above methodology on the following example. We consider an affine nonlinear system with actuator amplitude and rate limitations. The dynamic model of this nonlinear system is as follows [27]: x_ 1 ¼ ðx1 þ x22 Þ þ 10u1 þ sin2 ðx2 Þu2 þ 0:2d1 ðtÞ x_ 2 ¼ x21 þx21 u1 þ u2 þ 0:2d2 ðtÞ y1 ¼ x1 ,
y2 ¼ x2 ,
x1 ð0Þ ¼ 1,
x2 ð0Þ ¼ 0
ð56Þ
where f 1 ¼ ðx1 þ x22 Þ, f 2 ¼ x21 , g 11 ¼ 10, g 12 ¼ sin2 ðx2 Þ, g 21 ¼ x21 , g 22 ¼ 1. According to Theorem 2, the H1 tracking design is given as follows. We choose the following Gauss radial basis vector for approximating f1, i.e., 2
2
=b1
, . . . ,eJxc11 J
2
2
where e1 ¼ ½e1 11 , e2 ¼ ½e2 11 , e1 ¼ y1d y1 , e2 ¼ y2d y2 . The initial values for Hf i ð0Þ ði ¼ 1,2Þ and Hgij ð0Þ ði,j ¼ 1,2Þ are chosen as Hf 1 ð0Þ ¼ Hf 2 ð0Þ ¼ Hg 21 ð0Þ ¼ Hg 12 ð0Þ ¼ 0, Hg11 ð0Þ ¼ Hg 22 ð0Þ ¼ diagð1,1,1,1,1,1,1,1,1,1,1Þ. The auxiliary system is constructed as follows:
x_ 1 ¼ c1 x1 þ g^ 11 Du1 þ g^ 12 Du2 , x1 ð0Þ ¼ 0 x_ 2 ¼ c2 x2 þ g^ 21 Du1 þ g^ 22 Du2 , x2 ð0Þ ¼ 0 where c1 ¼ 25, c2 ¼ 25, Du1 ¼ u1 uc1 , Du2 ¼ u2 uc2 . We select the prescribed disturbance attenuation levels r1 ¼ r2 ¼ 0:5, and the supervisory control: ud1 ¼
1 T e P 1 B1 , 2r21 1
ud2 ¼
1 T e P2 B2 2r22 2
ð57Þ
According to (9), we can obtain 3# 0 2 T 3 " # 2 T Hg11 Ug 11 HTg12 Ug12 Hf 1 Uf 1 uc1 4 5 @ 4 5 ¼ uc ¼ uc2 HTg21 Ug 21 HTg22 Ug22 HTf2 Uf 2 " # " # " #! ud1 t1 25x1 þ x_ 1 þ þ ud2 t2 25x2 þ x_ 2
ð58Þ
and finally we get the dynamics of control u1 , u2 by assuming a first-order dynamics for uc1 , uc2 as u_ 1 ¼ sat10 ð20:5sat5 ðuc1 Þu1 Þ
where u1 , u2 are control inputs and have the limitations 9ui 9 r 5, 9u_ i 9 r 10, i ¼ 1,2, and d1 ðtÞ, d2 ðtÞ are bounded random noises in the interval ½0,1. It is desired to determine control the inputs u1 , u2 such that y1 , y2 follow those reference trajectories defined by y1d ¼ sinðtÞ, y2d ¼ cosðtÞ respectively. Rewrite the plant (56) as " # " # " #" # " # g 11 g 12 y_ 1 f1 u1 d1 ¼ þ þ 0:2 g 21 g 22 y_ 2 f2 u2 d2
Uf 1 ðxÞ ¼ ½eJxc1 J
_ g ¼ Gg Ug ðxÞBT P e u , H _ g ¼ Gg Ug ðxÞBT P e u H 1 1 1 1 1 1 1 2 11 11 11 12 12 12
=b11 T
u_ 2 ¼ sat10 ð20:5sat5 ðuc2 Þu2 Þ The MATLAB solver ‘‘ode4’’ is used to simulate the overall control system with step size 0.01. Simulation results are presented in Figs. 3–12. Fig. 3 shows the curves of output y1 ðtÞ and its reference trajectory, meanwhile Fig. 4 shows the curves of output y2 ðtÞ and its reference trajectory. Curves in Fig. 5 describe the tracking errors for y1 and y2. These curves indicate that the outputs track their reference values well, and the effects of approximation error and extern disturbance on tracking errors are effectively attenuated. The control signals u1 ðtÞ, u2 ðtÞ and their derivatives u_ 1 ðtÞ, u_ 2 ðtÞ are given in Figs. 6 and 7. It is observed that the control signals u1 ðtÞ and u2 ðtÞ satisfy the amplitude and rate limitations. Fig. 8 shows the signals uc1 ðtÞ, uc2 ðtÞ. Obviously they reference response
1
0.5
T
where x ¼ ðx1 ,x2 Þ , ci ði ¼ 1, . . . ,11Þ are the center of the radial base, and are chosen as c1 ¼ ð2,2ÞT , c2 ¼ ð1:6,1:6ÞT , c3 ¼ ð1:2,1:2ÞT , c4 ¼ ð0:8,0:8ÞT , c5 ¼ ð0:4,0:4ÞT , c6 ¼ ð0,0ÞT , c7 ¼ ð0:4,0:4ÞT , c8 ¼ ð0:8,0:8ÞT , c9 ¼ ð1:2,1:2ÞT , c10 ¼ ð1:6,1:6ÞT , c11 ¼ ð2,2ÞT , bi ¼ 2, i ¼ 1, . . . ,11. The radial bases for f 2 , g 11 , g 12 , g 21 , g 22 are chosen the same as f1. The robust control term s ¼ ðt1 , t2 ÞT ¼ ½y_ 1d þ l11 ðy1d y1 Þ, y_ 2d þ l21 ðy2d y2 ÞT and the coefficients l11 ¼ 5, l21 ¼ 5. Now we have A1 ¼ ½511 , A2 ¼ ½511 , B1 ¼ B2 ¼ ½111 , where ½11 represents a 1 1 matrix. Select Q 1 ¼ ½1011 and Q 2 ¼ ½1011 . Solving Lyapunov equations (51), we obtain P1 ¼ P2 ¼ ½111 . We choose the parameter update gain matrices for RBF neural
0
−0.5
−1 0
5
10
Fig. 3. The trajectory of y1.
15
20
78
R. Yuan et al. / Neurocomputing 125 (2014) 72–80
reference response
1
10 5
0.8
0
0.6
−5
0.4
−10
0.2 0 −0.2
10
−0.4
5
−0.6
0
5
10
15
20
0
5
10
15
20
0
−0.8
−5
−1
−10 0
5
10
15
20
Fig. 7. u_ 1 , u_ 2 .
Fig. 4. The trajectory of y2.
0.5 0 0 −5 −0.5 −1
−10 0
5
10
15
20
1.5
0
5
10
15
20
0
5
10
15
20
10
1 5
0.5 0 −0.5
0 0
5
10
15
20
Fig. 8. uc1 , uc2 .
Fig. 5. The tracking errors e1 , e2 .
10 5 0 0
5
10
15
20
0
5
10
15
20
0 −5 −10
Fig. 9. Du1 , Du2 .
Fig. 6. u1 , u2 .
do not satisfy the control input limitations. Fig. 9 shows the signals Du1 ðtÞ, Du2 ðtÞ. These curves in Fig. 9 show that Du1 ðtÞ, Du2 ðtÞ tend to zero soon as time goes. Fig. 10 shows the curves of x1 , x2 . From Figs. 9 and 10, we can see that x1 , x2 also tend to zero soon as Du1 ðtÞ, Du2 ðtÞ tend to zero. Figs. 11 and 12 show the tracking errors e1 , e2 at different prescribed disturbance attenuation levels r1 ¼ r2 ¼ 1,0:3,0:1. These curves indicate that under low disturbance attenuation level (ri is large, e.g. r1 ¼ r2 ¼ 1), the H1 tracking performance is poor than that under higher disturbance attenuation level (ri is small, i.e., r1 ¼ r2 ¼ 0:1).
In conclusion, the simulation results demonstrate that the adaptive controller proposed in preceding section not only can generate control signal that satisfies actuator amplitude and rate saturations but also can make sure that the closed loop system achieves H1 tracking performance.
5. Conclusions In this work, the control problem for a class of uncertain nonlinear MIMO systems with actuator amplitude and rate
R. Yuan et al. / Neurocomputing 125 (2014) 72–80
79
shows that the bound of tracking error is adjustable by an explicit choice of design parameters. The proposed controller can generate control signals satisfying their constraints and guarantee a desired closed loop performance. Simulation results illustrate the effectiveness of the proposed controller.
0.8 0.6 0.4 0.2 0 0
5
10
15
20
Acknowledgements
0 −0.2
This work is supported by National Natural Science Foundation (NNSF) of China under Grant 60904006.
−0.4 −0.6 −0.8 0
5
10
15
20
Fig. 10. x1 , x2 .
0.4 0.2 0 −0.2 0.06 −0.4
0.04 0.02
−0.6
0 −0.02
−0.8 −1
2
4
0
6
8
5
10
12
14
10
16
18
15
20 20
Fig. 11. The e1 at different disturbance attenuation levels.
1.2 1
0.15
0.8
0.05
0.1
0
0.6
−0.05 2
0.4
4
6
8
10
12
14
12
14
16
18
20
0.2 0 −0.2 −0.4 −0.6
0
2
4
6
8
10
16
18
20
Fig. 12. The e2 at different disturbance attenuation levels.
saturations is considered and an adaptive radial basis neural network based controller which is designed with a priori consideration of actuator saturation effects and guarantees H1 tracking performance is proposed. Adaptive radial basis function neural networks are used to approximate the unknown nonlinearities. An auxiliary system is constructed to compensate the effects of actuator amplitude and rate saturations. A supervisory control is designed to attenuate the effects of extern disturbance and neural network approximation errors, so that the closed loop system achieves a prescribed H1 tracking performance. Analysis
References [1] Jing Zhou, Changyun Wen, Adaptive backstepping control of uncertain systems, in: Lecture Notes in Control and Information Sciences, Springer Berlin, vol. 372, 2008, pp. 189–197. [2] Marious, Jay Farrell, Manu Sharma, On-line approximation control of uncertain nonlinear systems: issues with control input saturation, in: Proceedings of the 2003 American Control Conference, Colorado, 2003, pp. 543–548. [3] Jay Farrell, Marios Polycarpou, Manu Sharma, Adaptive backstepping with magnitude, rate, and bandwidth constraints: aircraft longitude control, in: Proceedings of the 2003 American Control Conference, Colorado, 2003, pp. 3898–3904. [4] Jay Farrell, Marios Polycarpou, Manu Sharma, On-line approximation based control of uncertain nonlinear systems with magnitude, rate and bandwidth constraints on the states and actuators, in: Proceedings of the 2004 American Control Conference, Massachusetts, 2004, pp. 2557–2562. [5] Jay Farrell, Manu Sharma, Marious Polycarpou, Backstepping based flight control with adaptive function approximation, J. Guid. Control Dyn. 28 (6) (2005) 1089–1102. [6] Jay A. Farrell, Marios Polycarpou, Manu Sharma, Wenjie Dong, Command filtered backstepping, IEEE Trans. Automat. Control 54 (6) (2009) 1391–1395. [7] Wenjie Dong, Jay A. Farrell, Marios M. Polycarpou, Command filtered adaptive backstepping, IEEE Trans. Control Syst. Technol. 20 (3) (2012) 566–580. [8] L. Sonneveldt, Q.P. Chu, J.A. Mulder, Nonlinear flight control design using constrained adaptive backstepping, J. Guid. Control Dyn. 30 (2) (2007) 322–336. [9] L. Sonneveldt, Q.P. Chu, J.A. Mulder, Constrained adaptive backstepping flight control: application to a nonlinear F-16/MATV model, In: 2006 AIAA Guidance, Navigation, and Control Conference and Exhibit, Keystone, Colorado, USA, 2006. [10] Changyuan Wen, Jing Zhou, Zhitao Liu, Hongye Su, Robust adaptive control of uncertain nonlinear system in the presence of input saturation and external disturbance, IEEE Trans. Automat. Control 56 (7) (2011) 1672–1678. [11] A.H. Bajodah, Asymptotic generalised dynamic inversion attitude control, IET Control Theory Appl. 4 (5) (2010) 827–840. [12] S. Siberling, A.P. Chu, J.A. Mulder, Robust flight control using incremental nonlinear dynamic inversion and angular acceleration prediction, J. Guid. Control Dyn. 33 (6) (2010) 1732–1742. [13] Monish Deepak Tandale, Adaptive Dynamic Inversion Control of Nonlinear Systems Subjected to Control Saturation Constraints, Doctor dissertation, Texas A&M University, Texas, USA, 2007. [14] Keisuke Enomoto, A study on the dynamic inversion method with nonlinear function considering saturation, in: AIAA Guidance, Navigation, and Control Conference, Toronto, Ontario Canada, 2010. [15] Eugene Lavretsky, Naira Hovakimyan, Positive m-modification for stable adaptation in dynamic inversion based adaptive control with input saturation, in: Proceedings of the 2005 American Control Conference, Portland, OR, USA, 2005, pp. 3373–3378. [16] Guido Hermann, Matthew C. Turner, Prathyush Menon, et al., Anti-windup synthesis for nonlinear dynamic inversion controllers, in: Proceedings of the IFAC ROCOND, Toulouse, France, 2006. [17] Prathyush P Menon, Guido Herrmann, Matthew Turner, et al., Nonlinear dynamic inversion based anti-windup: an aerospace application, in: Proceedings of the 17-th Word Congress of The International Federation of Automatic Control, Seoul, Korea, 2008, pp. 14156–14161. [18] Q. Hu, G.P. Rngaiah, Anti-windup schemes for uncertain nonlinear systems, IEE Proc. Control Theory Appl. 147 (3) (2000) 321–329. [19] G. Cybenko, Approximation by superpositions of a sigmoidal function, Math. Control Signals Syst. 2 (4) (1989) 303–314. [20] Simon Haykin, Neural Networks: A Comprehensive Foundation, 2nd Edition, Prentice-Hall Inc., 1999. [21] Eric N. Johnson, Anthony J. Calise, Hesham A. El-Shirbiny, Feedback linearization with neural network augmentation applied to X-33 attitude control, AIAA-2000-4157, 2000. [22] Chen mou, Zou Jie, Feng Xing, Jiang Changsheng, Approximation based tracking control of uncertain MIMO nonlinear systems with input saturation, in: Proceedings of the 29th Chinese Control Conference, China, 2010, pp. 6155–6160.
80
R. Yuan et al. / Neurocomputing 125 (2014) 72–80
[23] Yuto Yuasa, Noboru Sakamoto, Yoshio Umemura, Optimal control design for systems with input saturations and rate limiters, in: 2010 SICE Annual Conference, 2010, pp. 2042–2045. [24] Murad Abu-Khalaf, Frank L. Lewis, Nearly optimal control laws for nonlinear systems with saturating actuators using a neural network HJB approach, Automatica 41 (5) (2005) 779–791. ¨ ¨ [25] Lars Grune, Jurgen Pannek, Nonlinear Model Predictive Control: Theory and Algorithms, Springer, 2011. [26] Bing Xiao, Qinglei Hu, Adaptive sliding mode fault tolerant attitude tracking control for flexible spacecraft under actuator saturation, IEEE Trans. Control Syst. Technol. (99) (2011) 1–8. [27] Ruyi Yuan, Jianqiang Yi, Wensheng Yu, Guoliang Fan, Adaptive controller design for uncertain nonlinear systems with input magnitude and rate limitations, in: Proceedings of the 2011 American Control Conference, USA, 2011, pp. 3536–3541. [28] R. Penrose, A generalized inverse for matrices, Proc. Cambridge Philos. Soc. 51 (3) (1955) 406–413. [29] P. Ioannou, J. Sun, Robust Adaptive Control, Prentice-Hall Inc., 1996.
Ruyi Yuan is a research assistant in the Integrated Information System Research Center, Institute of Automation, Chinese Academy of Sciences. He received B.S. degree from Hunan University in 2006, Ph.D. from Institute of Automation, Chinese Academy of Sciences in 2011. His current research interest is in the area of nonlinear control, adaptive control and flight control.
Xiangmin Tan is a research assistant in the Integrated Information System Research Center, Institute of Automation Chinese Academy of Sciences, China. His research interests lie in the area of robotics, industrial control, fuzzy systems, and neural networks.
Guoliang Fan is a Senior Engineer at the Institute of Automation, Chinese Academy of Sciences. His research interest covers non-minimum phase system, nonlinear control, robust adaptive control, and flight control.
Jianqiang Yi received the B.E. degree from the Beijing Institute of Technology, Beijing, China, in 1985, and the M.E. and Ph.D. degrees from the Kyushu Institute of Technology, Kitakyushu, Japan, in 1989 and 1992 respectively. From 1992 to 1994, he was with the Computer Software Development Company, Tokyo, Japan. From 1994 to 2001, he was a Chief Researcher at MYCOM Inc., Kyoto, Japan. Currently, he is a Full Professor in the Institute of Automation, Chinese Academy of Sciences. His research interests include theories and applications of intelligent control, intelligent robotics, underactuated system control, slidingmode control, flight control, etc. He is an Associate Editor for the IEEE Computational Intelligence Magazine, Journal of Advanced Computational Intelligence and Intelligent Informatics, and Journal of Innovative Computing, Information and Control. He worked as a Research Fellow at CSD, Inc., Tokyo, Japan from 1992 to 1994, and a Chief Engineer at MYCOM, Inc., Kyoto, Japan from 1994 to 2001. Since 2001 he has been with the Institute of Automation, Chinese Academy of Sciences, China, where he is currently a Professor. His research interests lie in the area of intelligent control and robotics.
Contents lists available at ScienceDirect
Neurocomputing journal homepage: www.elsevier.com/locate/neucom
Robust adaptive neural network control for a class of uncertain nonlinear systems with actuator amplitude and rate saturations Ruyi Yuan n, Xiangmin Tan, Guoliang Fan, Jianqiang Yi Institute of Automation, Chinese Academy of Sciences, Beijing, PR China
a r t i c l e i n f o
a b s t r a c t
Available online 1 March 2013
An adaptive controller which is designed with a priori consideration of actuator saturation effects and guarantees H1 tracking performance for a class of multiple-input–multiple-output (MIMO) uncertain nonlinear systems with extern disturbances and actuator saturations is presented in this paper. Adaptive radial basis function (RBF) neural networks are used in this controller to approximate the unknown nonlinearities. An auxiliary system is constructed to compensate the effects of actuator saturations. Furthermore, in order to deal with approximation errors for unknown nonlinearities and extern disturbances, a supervisory control is designed, which guarantees that the closed loop system achieves a prescribed disturbance attenuation level so that H1 tracking performance is achieved. Steady and transient tracking performance are analyzed and the tracking error is adjustable by explicit choice of design parameters. Computer simulations are presented to illustrate the efficiency of the proposed controller. & 2013 Elsevier B.V. All rights reserved.
Keywords: Actuator saturation RBF neural network Adaptive control Robust control
1. Introduction All physical actuators in control systems have amplitude and rate limitations. For example, the elevator of an aircraft can only provide a limited force or torques in a limited rate. Actuator amplitude limitation or rate limitation constitutes a fundamental limitation on many linear or nonlinear control design techniques and has attracted the attention of numerous researchers. The controllers that ignore actuator limitations may cause the closed loop system performance to degenerate or even make the closed system unstable, and decrease the lifetime of the actuators, or damage the actuators. Higher performance may be expected if a controller is designed with a priori considering of the actuator saturation effects. The design of stabilizing controllers with a priori consideration of the actuator saturation effects for nonlinear systems with unknown nonlinearities and external disturbances is a challenging problem. Zhou [1] proposed an adaptive backstepping scheme to design an adaptive controller for a class of uncertain nonlinear single-input–single-output (SISO) systems in the presence of input saturations. To deal with saturations, an auxiliary system with the same order as that of the plant was constructed to compensate the effect of saturation. Farrell et al. [2–5] presented an adaptive backstepping approach and an online
n Correspondence to: Room 719, 95 Zhongguancun East Road, 100190 Beijing, PR China. Tel.: þ 86 10 62550985 16. E-mail addresses: [email protected] (R. Yuan), [email protected] (X. Tan), [email protected] (G. Fan), [email protected] (J. Yi).
0925-2312/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.neucom.2012.09.036
approximation based adaptive backstepping approach for unknown nonlinear systems with known magnitude, rate, bandwidth constraints on intermediate states or actuators without disturbance. Those approaches also used auxiliary systems for generating a modified tracking error to guarantee stability during saturation. Command filtered adaptive backstepping approaches [6–9] were also proposed to deal with the constraints on the control surfaces and the control states. For single input uncertain nonlinear systems in the presence of input saturation and unknown external disturbance, robust adaptive backstepping control algorithms were also developed by introducing a well defined smooth function and using a Nussbaum function which was used to compensate for the nonlinear term arising from the input saturation [10]. Dynamic inversion [11,12] approach is a widely used nonlinear control technique. However, the effects of actuator saturations have not been addressed with nominal dynamic inversion algorithm, so certain modifications are required. Tandale [13] proposed an adaptive dynamic inversion controller for a class of nonlinear systems with control saturation constraints. Enomoto [14] investigated the dynamic inversion control for nonlinear systems with control saturation constraints by Lyapunov synthesis. For a class of uncertain nonlinear dynamical systems in Brunovsky form, Lavretsky [15] proposed a dynamic inversion based adaptive control framework to provide stable adaptation in the presence of input constraints. The proposed design methodology can protect the control law from actuator position saturation. For a class of nonlinear systems which, in the presence of saturation, were controlled by nonlinear dynamic inversion
R. Yuan et al. / Neurocomputing 125 (2014) 72–80
controllers, an anti-windup compensation scheme was also proposed [16–18]. Neural network technique [19,20] was also used to handle actuator saturations problem. Calise [21] introduced a neural network based method termed as Pseudo-Control Hedging (PCH) for addressing a wide class of plant input characteristics such as actuator position limits, actuator rate limits, time delay, and input quantization. Chen et al. [22] also introduced a radial basis function neural network based controller for uncertain MIMO nonlinear systems with input saturations. The control design for nonlinear systems with actuator saturations was also be investigated by optimal control [23], nearly optimal control [24], nonlinear model predictive control [25] and fault tolerant scheme [26], etc. However, there are still few results for the control of uncertain nonlinear systems by taking actuator saturations into account in the controller design and analysis. In [27], the authors proposed an adaptive controller for MIMO nonlinear systems with control input limitations by using an auxiliary system and extended tracking errors which were used in neural network parameter update laws to compensate the effects of control input limitations. In this paper, for the control of a class of MIMO uncertain nonlinear systems in the presence of disturbances and actuator saturations, dynamic inversion [11] based controller which can generate constrained control signal is designed. Adaptive RBF neural networks are used to approximate unknown nonlinearities. An auxiliary system is constructed to compensate the effects of actuator amplitude and rate saturations. This auxiliary system and compensation scheme are different from [27], so that the extended tracking error in [27] is no longer needed. A supervisory control is designed to attenuate the effects of approximation errors and external disturbance so as to guarantee a H1 tracking performance. The performance of the closed loop system is obtained through Lyapunov analysis. The bounds of tracking errors can be adjusted by tuning the design parameters. The proposed controller can generate control signals satisfying actuator amplitude and rate limitations, and guarantee a H1 tracking performance of the closed loop system. The rest of this paper is organized as follows. In Section 2, the problem statement is presented. In Section 3, the adaptive control scheme is discussed, and the closed loop system performance is analyzed. A numerical example is shown in Section 4. Section 5 concludes the paper. Throughout this paper, 9 9 indicates the absolute value, J J indicates the Euclidean vector norm, and J J2 indicates the L2 norm.
2. Problem formulation Consider the class of MIMO systems described by the following differential equations: x_ i1 ¼ xi2 ^ x_ i,ri 1 ¼ xiri x_ iri ¼ f i ðxÞ þ
m X
ð1Þ g ij ðxÞuj þ di
i ¼ 1, . . . ,m
yðnÞ ¼ FðxÞ þGðxÞu þ d
ð2Þ m
ðnÞ def
..., where y ¼ ½y1 , . . . ,ym T A R is the output vector; y ¼ Pm ri ri ðr i Þ m mÞ T yðr r ¼ n; y ¼ d y =dt ; x ¼ ½x , . . . ,x , ..., 11 1r i m AR , i i¼1 1 i xm1 , . . . ,xmrm T A Rn is the state vector available for measurement; u ¼ ½u1 , . . . ,um A Rm is the control vector with u_ i rvimax
3. Design of adaptive controllers To begin, define t1 , . . . , tm as follows:
ti ¼ yðridi Þ þ
ri X
lij eðj1Þ , i ¼ 1, . . . ,m i
j¼1
i ¼ 1, . . . ,m are the reference signals, where yid , ei ¼ yid yi ði ¼ 1, . . . ,mÞ are the tracking errors, li1 , . . . , liri are parameters which make sure that the roots of the equation sri þ liri sri 1 þ þ li2 s þ li1 ¼ 0 are all in the open left-half complex plane. If FðxÞ and GðxÞ are known and the constrains on control inputs are ignored, then based on dynamic inversion algorithm, the control law: uc0 ¼ G1 ðxÞðFðxÞ þ sÞ
ð4Þ
can be applied to (2) to achieve the following asymptotically stable tracking: 2 3 r1 X ðr1 Þ ðj1Þ þ l e e 1j 1 6 1 7 6 7 j¼1 6 7 6 7 ^ ð5Þ 6 7¼0 6 7 r m 6 ðrm Þ X 7 ðj1Þ 5 4 em þ lmj em j¼1
in the case of no external disturbances. Because FðxÞ and GðxÞ are unknown vector and matrix respectively, the above control law (4) cannot be implemented in practice. Besides, there is no guarantee that uc0 satisfies the actuator constraints (3). It is well known that neural networks [19,20] can be used as universal approximators to approximate any continuous functions at any arbitrary accuracy as long as the network is big enough. In this work, in order to treat this tracking control design problem, radial basis function (RBF) neural networks are used to approximate the unknown functions, that is, f i ðxÞ, i ¼ 1, . . . ,m, and g ij ðxÞ, i,j ¼ 1, . . . ,m are approximated as follows: f i ðxÞ f^ i ðx9Hf i Þ ¼ HTfi Uf i ðxÞ,
i ¼ 1, . . . ,m i, j ¼ 1, . . . ,m
ð6Þ ð7Þ
M
which also can be rewritten in the following compact form:
9ui 9 r uimax ,
where uimax and vimax denote the actuator amplitude and rate limits respectively. FðxÞ ¼ ½f 1 ðxÞ, . . . ,f m ðxÞ A Rm , GðxÞ ¼ ½g ij ðxÞmm A Rmm (½mm represents a m m matrix) are continuous unknown functions of the state x. di denotes the external disturbances which RT 2 is unknown but bounded and satisfies d¼ 0 di dt o 1. m ½d1 , . . . ,dm A R . The control objective is to force yi to follow a given bounded reference signal yid in the presence of actuator saturations and extern disturbances. For (2) to be controllable, we assume that sðGðxÞÞ a0 for x in certain controllability region Uc A Rn , where sðGðxÞÞ denotes the minimum singular value of the matrix GðxÞ.
g ij ðxÞ g^ ij ðx9Hgij Þ ¼ HTgij Ugij ðxÞ,
j¼1
yi ¼ xi1 ,
73
½y1ðr1 Þ ,
ð3Þ
where Hf i A RMf i , Hgij A R gij are weight vectors, and Uf i ðxÞ A M RMf i ðxÞ, Ug ij ðxÞ A R gij ðxÞ are radial basis vectors, M f i , M g ij are the corresponding dimensions of the basis vectors. Denote 2 3 2 3 g^ 11 ðxÞ g^ 1m ðxÞ f^ 1 ðxÞ 6 7 6 7 ^ ^ HF Þ ¼ 6 ^ 7, Gðx9 & ^ Fðx9 ð8Þ HG Þ ¼ 4 ^ 5 4 5 ^f ðxÞ ^ m1 ðxÞ g^ mm ðxÞ g m ^ ^ HF Þ is an estimation of FðxÞ, and Gðx9 Then Fðx9 HG Þ is an estimation of GðxÞ.
74
R. Yuan et al. / Neurocomputing 125 (2014) 72–80
Using the approximation (8), and considering the control inputs constraints and extern disturbances, we modify the control law (4) as follows: #
uc ¼ G^ ðx9HG ÞðF^ ðx9HF Þ þ s þ g þ ud Þ
This means that by choosing appropriate parameters oi , zi , not only the actuators amplitude and rate saturations but also the dynamics of the actuators, like dampings and frequencies, can be integrated into the controller.
ð9Þ Remark 1. We known that for the first-order system:
u ¼ satðuc Þ
ð10Þ
g ¼ Cnn_
ð11Þ
where satðuc Þ represents the amplitude and rate limitations on uc . s ¼ ðt1 , . . . , tm ÞT is the robust control term. ud A Rm , which will be determined later, is a supervisor control used to attenuate the extern disturbance d. G# ðxÞ represents the generalized inversion [28] of GðxÞ. n ¼ ðx1 , . . . , xm ÞT A Rm is the state of the following constructed auxiliary system (12) which uses the difference of uc before and after amplitude and rate limitations as input: ^ n_ ¼ Cn þ Gðx9 HG ÞDu
ð12Þ
C ¼ diagðc1 , . . . ,cm Þ A Rmm , ci ði ¼ 1, . . . ,mÞ are positive parameters, Du ¼ uuc . n is used to compensate the effect of actuator saturations. uc is obtained according to certainty equivalence principle [29] which is widely used in adaptive control schemes. However, there is also no guarantee that uc satisfies the constraints (3), hence amplitude and rate limitations satðuc Þ are imposed on uc to generate u which satisfies the constraints (3). The amplitude and rate limitations on uc , i.e., satðuc Þ, can always be implemented by assuming a first-order model or second-order model for the dynamics of each component of uc , for example, u_ i ¼ satR ðoi ðsatA ðuci Þui ÞÞ or u€ i ¼ satR ðo2i ðsatA ðuci Þui Þ2zi oi u_ i Þ, where ui , uci are the i-th elements of u, uc respectively, oi , zi are positive constants, satR ðÞ, satA ðÞ represent the rate and amplitude saturation functions respectively. The function satR ðxÞ is defined as follows: 8 if x Z R > : R if x r R and satA ðxÞ is defined similarly. Fig. 1 gives visual descriptions for the first-order and second-order models respectively. In the linear range of the function satA ðxÞ and satR ðxÞ, the transfer function for the first-order model is Ui oi ¼ U ci s þ oi
ð13Þ
x_ ¼ f ðxÞ þ u
ð15Þ
if f ðxÞ is known, then we can construct a new system _ ðvÞ u ¼ vf
ð16Þ
with v as input and u as output. Eqs. (15) and (16) yield a closedloop system: _ ðxÞ ¼ vf _ ðvÞ xf
For system (17), the output x is totaly the same as the input v if and only if xð0Þ ¼ vð0Þ. The transform function from v to x is 1. Actually, the zeros of (16) is the poles of (15), and (16) is the zero assignment based inverse system of (15). Similarly, for the ^ auxiliary system (12), if we view the component Gðx9 HG ÞDu as the control input, then the new system: ^ HG ÞDu ¼ Ck þ k_ Gðx9
ð18Þ
^ HG ÞDu as the output is the zero with k as the input and Gðx9 assignment based inverse system of (12), (18) and (12) yield a closed loop system:
n_ þ Cn ¼ k_ þCk
ð19Þ
The output n is totaly the same as the input k if and only if nð0Þ ¼ kð0Þ. So n_ þ Cn or k_ þ Ck can be used as a prediction of ^ HG ÞDu which is the effect of actuator saturations and can be Gðx9 used to compensate the effect of actuator saturations. From (9)–(12), we have #
u ¼ satðuc Þ ¼ G^ ðx9HG ÞðF^ ðx9HF Þ þ s þ ud Þ
Remark 2. From (12) we also have that 0 1 Z tX m ci t @ c n i g^ ij ðxðnÞÞDuj ðnÞe dnA xi ðtÞ ¼ e xð0Þ þ 0 j¼1
ð14Þ
ð20Þ
By ignoring the term ud , u which satisfies the constraints (3) has the same form as uc0 which is obtained by dynamic inversion algorithm [11], which makes the controller be a adaptive neural network based dynamic inversion controller and the perfect tracking be available.
and the transfer function for the second model is
o2i Ui ¼ 2 U ci s þ 2zi oi s þ o2i
ð17Þ
¼ eci t xð0Þ þ
Z
t
m X
0 j¼1
g^ ij ðxðnÞÞDuj ðnÞeci ðtnÞ dn
Fig. 1. Schematic for amplitude and rate limitations. (a) First-order model and (b) second-order model.
R. Yuan et al. / Neurocomputing 125 (2014) 72–80
r eci t 9xð0Þ9 þ
Z
t
m X
0 j¼1
r eci t 9xð0Þ9 þ
m ci
9g^ ij ðxðnÞÞ9Duj ðnÞ9Þeci ðtnÞ dn
sup 9g^ ij ðxðnÞÞ9 sup 9Duj ðnÞ9
0rnrt
ð21Þ
0rnrt
where Duj ðnÞ ¼ uj ðnÞucj ðnÞ. Because ci is positive, the auxiliary system is bounded input bounded output stable. Remark 3. Although the true value of GðxÞ are invertible accord^ ing to the assumption, the estimate matrix Gðx9 HG Þ may become singular during the adaptive process, so Moore–Penrose general^ ized matrix inverse [28] of Gðx9 HG Þ is used. Remark 4. In the case of known FðxÞ and GðxÞ and free of external disturbances, the control low
75
where udi , di ,wi are the i-th element of ud , d, and w respectively. ~ ¼ H Hn , H ~ g ¼ Hg Hn , then Defining ei ¼ ½ei , . . . ,eiðri 1Þ T , H fi fi fi g ij ij ij Eq. (31) can be rewritten in the following form: 0 1 m X T T ~ ~ e_ i ¼ Ai ei þ Bi @H f i Uf i ðxÞ þ ð32Þ ðH g ij Ug ij ðxÞÞuj udi di þ wi A j¼1
where 2
0 6 0 6 6 ^ Ai ¼ 6 6 6 0 4 li1
1
0
0
0
1
^
^
&
0
0
li2
li3
3
0 7 7 7 ^ 7, 7 1 7 5 liri
2 3 0 607 6 7 6 7 7 Bi ¼ 6 6^7 6 7 405 1
ð22Þ
For the i-th subsystem of (26), the following theorem can be obtained.
uc ¼ G1 ðxÞðFðxÞ þ s þ gÞ
ð23Þ
Theorem 1. For the i-th subsystem of (26), if we select the control law (10), and the following parameters update laws and udi
g ¼ Cnn_
ð24Þ
_ ¼ G U ðxÞBT P e H fi fi fi i i i
ð33Þ
n_ ¼ Cn þ GðxÞDu
ð25Þ
_ g ¼ Gg Ug ðxÞBT P e u H i i i j ij ij ij
ð34Þ
u ¼ satðuc Þ with
can be applied to System (2) to obtain the asymptotically stable tracking (5).
udi ¼
In the following, we will specify the update laws for the RBF parameters Hf i ði ¼ 1, . . . ,mÞ, Hg ij ði,j ¼ 1, . . . ,mÞ and the supervisor control ud , so that desired tracking performance can be achieved. Applying the control law (9) and (10) to System (2) yields
then the following H1 tracking performance can be obtained: Z T ~ ð0Þ ~ T ð0ÞG1 H eTi Q i ei dt r eTi ð0ÞPi ei ð0Þ þ H fi fi fi 0 Z m T X T ~ ð0ÞG1 H ~ g ð0Þ þ r2 H R2i dt ð36Þ þ g ij g ij i ij
^ ^ HF ÞFðxÞ þðGðx9 sYðrÞ ¼ Fðx9 HG ÞGðxÞÞuud d Define the optimal approximation weight f i ði ¼ 1, . . . ,mÞ, g ij ði,j ¼ 1, . . . ,mÞ as follows: " # Hn ¼ arg min sup 9f ðxÞf^ ðx9H Þ9 fi
i
Hf i A OF x A Uc
i
fi
" n
Hg ij ¼ arg min
Hgij A OG
for
ð27Þ
sup 9g ij ðxÞg^ ij ðx9Hgij Þ9
ð28Þ
x A Uc
According to universal approximation property of neural networks [19,20], the following assumption is reasonable: Assumption 1. The minimum approximation error is square integrable, i.e., Z T wT w dt o 1 ð30Þ 0
Using the optimal approximation for FðxÞ, GðxÞ, the i-th subsystem of (26) can be rewritten as ri X
lik eðk1Þ ¼ f^ i ðx9Hf i Þf^ i ðx9Hnf i Þudi di þ wi i
k¼1
þ
m X j¼1
ð35Þ
0
where Gf i , Ggij ðj ¼ 1, . . . ,mÞ are positive definite diagonal matrices,
ri ði ¼ 1, . . . ,mÞ are positive parameters representing for prescribed def disturbance attenuation levels, Ri ¼ di þ wi , Q i A Rmm is arbitrary
symmetric positive definite matrices, Pi A Rmm is the symmetric positive definite solution of the following Lyapunov equation: Pi Ai þ ATi Pi ¼ Q i
#
where OF , OG , Uc denote the sets of suitable bounds on Hf i , Hg ij , and x respectively. Hnf i ði ¼ 1, . . . ,mÞ, Hngij ði,j ¼ 1, . . . ,mÞ are constant vectors. The optimal approximations for FðxÞ and GðxÞ are ^ ^ Hn Þ, Gðx9 denoted as Fðx9 HnG Þ respectively. Define the minimum F approximation error as h i def ^ ^ w ¼ Fðx9 HnF ÞFðxÞ þ Gðx9 HnG ÞGðxÞ u ð29Þ
iÞ eðr i þ
j¼1
ð26Þ vectors
1 T e P i Bi 2r2i i
n
ðg^ ij ðx9Hgij Þg^ ij ðx9Hgij ÞÞuj
ð31Þ
ð37Þ
Proof. Define the Lyapunov function Vi for the i-th subsystem as follows: Vi ¼
m 1 T 1 ~ T 1 ~ 1X ~ T G1 H ~g ei Pi ei þ H H f i Gf i H f i þ ij 2 2 2 j ¼ 1 gij gij
ð38Þ
The time derivative of Vi is m X 1 T ~_ þ ~_ g ~ T G1 H ~ T G1 H H V_ i ¼ ðe_ i Pi ei þ eTi Pi e_ i Þ þ H fi fi fi g ij g ij ij 2 j¼1
¼
1 T T ~ T UF ðxÞBT P e ðe A P e þ eTi Pi Ai ei Þ þ H i i i fi 2 i i i i m m X X ~ T U ðxÞBT P e u þ H ~ T G1 H ~_ þ ~_ g ~ T G1 H þ H H G fi g ij fi fi g ij g ij i i i j ij i¼1
j¼1
1 þ ðudi þ Ri ÞðBTi Pi ei þeTi Pi Bi Þ 2 1 1 1 r eTi Qei þ Ri ðBTi Pi ei þ eTi Pi Bi Þ 2 eTi Pi Bi BTi Pi ei 2 2 2r 2 1 1 1 1 r eTi Qei þ r2i R2i ri Ri eTi Pi Bi 2 2 2 ri 1 T 1 2 2 r ei Qei þ ri Ri 2 2
ð39Þ
Integrating both sides of the above inequality from 0 to T yields Z Z r2 T 2 1 T T ei Q i ei dt rV i ð0ÞV i ðTÞ þ i R dt ð40Þ 2 0 2 0 i
76
R. Yuan et al. / Neurocomputing 125 (2014) 72–80
Since VðTÞ 4 0, the inequality (40) implies the following inequality: Z Z r2 T 2 1 T T ei Q i ei dt rV i ð0Þ þ i R dt ð41Þ 2 0 2 0 i According to the definition of Vi, the following inequality is obtained: Z T ~ ð0Þ ~ T ð0ÞG1 H eTi Q i ei dt r eTi ð0ÞPi ei ð0Þ þ H fi fi fi 0
þ
m X
T
~ ð0ÞG1 H ~ g ð0Þ þ r2 H g ij g ij i ij
j¼1
This is (36).
Z
T 0
R2i dt
ð42Þ
&
Finally, for the nonlinear system (2), the following theorem can be obtained.
definite solutions of the following Lyapunov equations: Pi Ai þ ATi Pi ¼ Q i ,
A scheme of the adaptive H Fig. 2.
ð43Þ
1
ð51Þ
tracking control scheme is shown in
P Proof. Define the Lyapunov function V ¼ m i ¼ 1 V i where Vi is defined in (38). According to the definition of Vi and Theorem 1, it is easy to obtain (50). & Corollary 1. The closed loop system is stable and the steady tracking errors satisfy limt-1 ei ¼ 0,i ¼ 1, . . . ,m, i.e., limt-1 9yid ðtÞyi ðtÞ9 ¼ 0. The bound of the transient tracking errors will be given by Jei J22 r
Theorem 2. For the nonlinear system (2), if we select the control law u ¼ satðuc Þ
i ¼ 1, . . . ,m
~ ð0ÞÞ ~ T ð0ÞG1 H 2ðeTi ð0ÞPi ei ð0Þ þ H fi fi fi
lmin ðQ i Þ 2
R 1 ~ 2 T 2 ~T j ¼ 1 H g ij ð0ÞGg ij H g ij ð0Þ þ ri 0 Ri dt
Pm
þ
lmin ðQ i Þ
ð52Þ
where lmin ðQ i Þ represents the minimum eigenvalue of matrix Q i .
with Proof. From (39), it can be obtained that
#
uc ¼ G^ ðx9HG ÞðF^ ðx9HF Þ þ s þ g þ ud Þ
ð44Þ
g ¼ Cnn_
ð45Þ
^ n_ ¼ Cn þ Gðx9 HG ÞDu
ð46Þ
_ ¼ G U ðxÞBT P e , i ¼ 1, . . . ,m H fi fi fi i i i
ð47Þ
e2i rJei J2 r
_ g ¼ Gg Ug ðxÞBT P e u , i,j ¼ 1, . . . ,m H i i i j ij ij ij
ð48Þ
hence
1 T e Pi Bi , udi ¼ 2r2i i
i ¼ 1, . . . ,m
ð49Þ
then the following H1 tracking performance can be obtained: Z T m X ~ T ð0ÞG1 H ~ ð0Þ eT Qe dt r eT ð0ÞPeð0Þ þ H fi fi fi 0
i¼1
þ
m X i,j ¼ 1
~ T ð0ÞG1 H ~ g ð0Þ þ H g ij g ij ij
m X i¼1
r2i
Z 0
T
R2i dt
ð50Þ
where Gf i ði ¼ 1, . . . ,mÞ, Gg ij ði,j ¼ 1, . . . ,mÞ are positive definite diagonal matrices, ri ði ¼ 1, . . . ,mÞ are positive parameters representing for prescribed disturbance attenuation levels, e ¼ ½eT1 , . . . ,eTm T , Ri def ¼ di þ wi , Q ¼ diagðQ 1 , . . . ,Q m Þ and Q i A Rmm ði ¼ 1, . . . ,mÞ are arbitrary symmetric positive definite matrices, P ¼ diag ðP1 , . . . ,Pm Þ and Pi ði ¼ 1, . . . ,mÞ are the symmetric positive
Δ
V_i r 12 lmin ðQ i ÞJei J2 þ 12r2i R2i
ð53Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi _ the trackV i is negative whenever Jei J Z ri 9Ri 9= lmin ðQ i Þ. Hence pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ing error will stay in the region fei j Jei J r ri 9Ri 9= lmin ðQ i Þg. Obviously 2V_i þ r2i R2i lmin ðQ i Þ
ð54Þ
RT 2ðV i ð0ÞV i ðTÞÞ þ r2i 0 R2i dt lmin ðQ i Þ 0 R 2 T 2 2V i ð0Þ þ ri 0 Ri dt r lmin ðQ i Þ 2 ~ ð0Þ ~ T ð0ÞG1 H ðeT ð0ÞPi ei ð0Þ þ H ¼ fi fi fi lmin ðQ i Þ i R P 1 ~ 1 2 T 2 ~T 2ð m j ¼ 1 H g ij ð0ÞGg ij H g ij ð0Þ þ 2 ri 0 Ri dtÞ ð55Þ þ lmin ðQ i Þ RT 2 RT 2 then Assumption 1 implies 0 wi dt o 1, 0 Ri dt ¼ RT 2 0 ðwi di Þ dt o 1. Eq. (55) means ei A L2 . According to Barbalat lemma, limt-1 ei ¼ 0. & Jei J22 ¼
Z
T
e2i dt r
Remark 5. According to Theorem 1, the i-th subsystem achieves a H1 tracking performance with a prescribed disturbance
Δ
Fig. 2. The overall structure of adaptive H1 tracking control scheme.
R. Yuan et al. / Neurocomputing 125 (2014) 72–80
77
attenuation level ri , i.e., the L2 gain from wi to the tracking error ei is equal or less than ri .
networks as Gf 1 ¼ Gf 2 ¼ ¼ Gg11 ¼ Gg12 ¼ Gg21 ¼ Gg 22 ¼ diagð1,1, 1,1,1,1,1,1,1,1,1Þ and set the parameters update laws as follows:
Remark 6. Because the parameters lij , j ¼ 1, . . . ,r i can make Ai be a Hurwitz stable matrix, there exists unique symmetric positive definite matrix Pi satisfying Lyapunov equation (37).
_ ¼ G U ðxÞBT P e _ ¼ G U ðxÞBT P e , H H f1 f1 f1 f2 f2 f2 1 1 1 2 2 2
Remark 7. The bound for Jyid ðtÞyi ðtÞJ2 is an explicit function of the design parameters. According to Corollary 1, this bound ~ ð0Þ, depends on the initial estimate errors H fi ~ g ð0Þ ðj ¼ 1, . . . ,mÞ. The closer the initial estimates H ð0Þ, H fi ij Hg ij ð0Þ ðj ¼ 1, . . . ,mÞ to the true values Hnf i ð0Þ, Hngij ð0Þ ðj ¼ 1, . . . ,mÞ, the better the transient performance. The effects of initial estimate errors on this bound can be decreased by increasing the values of the diagonal adaptation gain matrices Gf i , Ggij ðj ¼ 1, . . . ,mÞ and by choosing positive definite symmetric matrix Q i with larger minimum eigenvalue. On the other hand, this bound also depends on the external disturbances and neural network approximation errors. The effects of extern disturbances and neural network approximation errors on the transient performance can be reduced by decreasing ri and increasing lmin ðQ i Þ. Large ri represents low disturbance attenuation level while small ri represents high disturbance attenuation level.
_ g ¼ Gg Ug ðxÞBT P e u , H _ g ¼ Gg Ug ðxÞBT P e u H 2 2 2 1 2 2 2 2 21 21 21 22 22 22
Remark 8. If Du ¼ 0 or Du tends to zero as t tends to infinity, then xi -0. 4. Numerical example In this section, we illustrate the above methodology on the following example. We consider an affine nonlinear system with actuator amplitude and rate limitations. The dynamic model of this nonlinear system is as follows [27]: x_ 1 ¼ ðx1 þ x22 Þ þ 10u1 þ sin2 ðx2 Þu2 þ 0:2d1 ðtÞ x_ 2 ¼ x21 þx21 u1 þ u2 þ 0:2d2 ðtÞ y1 ¼ x1 ,
y2 ¼ x2 ,
x1 ð0Þ ¼ 1,
x2 ð0Þ ¼ 0
ð56Þ
where f 1 ¼ ðx1 þ x22 Þ, f 2 ¼ x21 , g 11 ¼ 10, g 12 ¼ sin2 ðx2 Þ, g 21 ¼ x21 , g 22 ¼ 1. According to Theorem 2, the H1 tracking design is given as follows. We choose the following Gauss radial basis vector for approximating f1, i.e., 2
2
=b1
, . . . ,eJxc11 J
2
2
where e1 ¼ ½e1 11 , e2 ¼ ½e2 11 , e1 ¼ y1d y1 , e2 ¼ y2d y2 . The initial values for Hf i ð0Þ ði ¼ 1,2Þ and Hgij ð0Þ ði,j ¼ 1,2Þ are chosen as Hf 1 ð0Þ ¼ Hf 2 ð0Þ ¼ Hg 21 ð0Þ ¼ Hg 12 ð0Þ ¼ 0, Hg11 ð0Þ ¼ Hg 22 ð0Þ ¼ diagð1,1,1,1,1,1,1,1,1,1,1Þ. The auxiliary system is constructed as follows:
x_ 1 ¼ c1 x1 þ g^ 11 Du1 þ g^ 12 Du2 , x1 ð0Þ ¼ 0 x_ 2 ¼ c2 x2 þ g^ 21 Du1 þ g^ 22 Du2 , x2 ð0Þ ¼ 0 where c1 ¼ 25, c2 ¼ 25, Du1 ¼ u1 uc1 , Du2 ¼ u2 uc2 . We select the prescribed disturbance attenuation levels r1 ¼ r2 ¼ 0:5, and the supervisory control: ud1 ¼
1 T e P 1 B1 , 2r21 1
ud2 ¼
1 T e P2 B2 2r22 2
ð57Þ
According to (9), we can obtain 3# 0 2 T 3 " # 2 T Hg11 Ug 11 HTg12 Ug12 Hf 1 Uf 1 uc1 4 5 @ 4 5 ¼ uc ¼ uc2 HTg21 Ug 21 HTg22 Ug22 HTf2 Uf 2 " # " # " #! ud1 t1 25x1 þ x_ 1 þ þ ud2 t2 25x2 þ x_ 2
ð58Þ
and finally we get the dynamics of control u1 , u2 by assuming a first-order dynamics for uc1 , uc2 as u_ 1 ¼ sat10 ð20:5sat5 ðuc1 Þu1 Þ
where u1 , u2 are control inputs and have the limitations 9ui 9 r 5, 9u_ i 9 r 10, i ¼ 1,2, and d1 ðtÞ, d2 ðtÞ are bounded random noises in the interval ½0,1. It is desired to determine control the inputs u1 , u2 such that y1 , y2 follow those reference trajectories defined by y1d ¼ sinðtÞ, y2d ¼ cosðtÞ respectively. Rewrite the plant (56) as " # " # " #" # " # g 11 g 12 y_ 1 f1 u1 d1 ¼ þ þ 0:2 g 21 g 22 y_ 2 f2 u2 d2
Uf 1 ðxÞ ¼ ½eJxc1 J
_ g ¼ Gg Ug ðxÞBT P e u , H _ g ¼ Gg Ug ðxÞBT P e u H 1 1 1 1 1 1 1 2 11 11 11 12 12 12
=b11 T
u_ 2 ¼ sat10 ð20:5sat5 ðuc2 Þu2 Þ The MATLAB solver ‘‘ode4’’ is used to simulate the overall control system with step size 0.01. Simulation results are presented in Figs. 3–12. Fig. 3 shows the curves of output y1 ðtÞ and its reference trajectory, meanwhile Fig. 4 shows the curves of output y2 ðtÞ and its reference trajectory. Curves in Fig. 5 describe the tracking errors for y1 and y2. These curves indicate that the outputs track their reference values well, and the effects of approximation error and extern disturbance on tracking errors are effectively attenuated. The control signals u1 ðtÞ, u2 ðtÞ and their derivatives u_ 1 ðtÞ, u_ 2 ðtÞ are given in Figs. 6 and 7. It is observed that the control signals u1 ðtÞ and u2 ðtÞ satisfy the amplitude and rate limitations. Fig. 8 shows the signals uc1 ðtÞ, uc2 ðtÞ. Obviously they reference response
1
0.5
T
where x ¼ ðx1 ,x2 Þ , ci ði ¼ 1, . . . ,11Þ are the center of the radial base, and are chosen as c1 ¼ ð2,2ÞT , c2 ¼ ð1:6,1:6ÞT , c3 ¼ ð1:2,1:2ÞT , c4 ¼ ð0:8,0:8ÞT , c5 ¼ ð0:4,0:4ÞT , c6 ¼ ð0,0ÞT , c7 ¼ ð0:4,0:4ÞT , c8 ¼ ð0:8,0:8ÞT , c9 ¼ ð1:2,1:2ÞT , c10 ¼ ð1:6,1:6ÞT , c11 ¼ ð2,2ÞT , bi ¼ 2, i ¼ 1, . . . ,11. The radial bases for f 2 , g 11 , g 12 , g 21 , g 22 are chosen the same as f1. The robust control term s ¼ ðt1 , t2 ÞT ¼ ½y_ 1d þ l11 ðy1d y1 Þ, y_ 2d þ l21 ðy2d y2 ÞT and the coefficients l11 ¼ 5, l21 ¼ 5. Now we have A1 ¼ ½511 , A2 ¼ ½511 , B1 ¼ B2 ¼ ½111 , where ½11 represents a 1 1 matrix. Select Q 1 ¼ ½1011 and Q 2 ¼ ½1011 . Solving Lyapunov equations (51), we obtain P1 ¼ P2 ¼ ½111 . We choose the parameter update gain matrices for RBF neural
0
−0.5
−1 0
5
10
Fig. 3. The trajectory of y1.
15
20
78
R. Yuan et al. / Neurocomputing 125 (2014) 72–80
reference response
1
10 5
0.8
0
0.6
−5
0.4
−10
0.2 0 −0.2
10
−0.4
5
−0.6
0
5
10
15
20
0
5
10
15
20
0
−0.8
−5
−1
−10 0
5
10
15
20
Fig. 7. u_ 1 , u_ 2 .
Fig. 4. The trajectory of y2.
0.5 0 0 −5 −0.5 −1
−10 0
5
10
15
20
1.5
0
5
10
15
20
0
5
10
15
20
10
1 5
0.5 0 −0.5
0 0
5
10
15
20
Fig. 8. uc1 , uc2 .
Fig. 5. The tracking errors e1 , e2 .
10 5 0 0
5
10
15
20
0
5
10
15
20
0 −5 −10
Fig. 9. Du1 , Du2 .
Fig. 6. u1 , u2 .
do not satisfy the control input limitations. Fig. 9 shows the signals Du1 ðtÞ, Du2 ðtÞ. These curves in Fig. 9 show that Du1 ðtÞ, Du2 ðtÞ tend to zero soon as time goes. Fig. 10 shows the curves of x1 , x2 . From Figs. 9 and 10, we can see that x1 , x2 also tend to zero soon as Du1 ðtÞ, Du2 ðtÞ tend to zero. Figs. 11 and 12 show the tracking errors e1 , e2 at different prescribed disturbance attenuation levels r1 ¼ r2 ¼ 1,0:3,0:1. These curves indicate that under low disturbance attenuation level (ri is large, e.g. r1 ¼ r2 ¼ 1), the H1 tracking performance is poor than that under higher disturbance attenuation level (ri is small, i.e., r1 ¼ r2 ¼ 0:1).
In conclusion, the simulation results demonstrate that the adaptive controller proposed in preceding section not only can generate control signal that satisfies actuator amplitude and rate saturations but also can make sure that the closed loop system achieves H1 tracking performance.
5. Conclusions In this work, the control problem for a class of uncertain nonlinear MIMO systems with actuator amplitude and rate
R. Yuan et al. / Neurocomputing 125 (2014) 72–80
79
shows that the bound of tracking error is adjustable by an explicit choice of design parameters. The proposed controller can generate control signals satisfying their constraints and guarantee a desired closed loop performance. Simulation results illustrate the effectiveness of the proposed controller.
0.8 0.6 0.4 0.2 0 0
5
10
15
20
Acknowledgements
0 −0.2
This work is supported by National Natural Science Foundation (NNSF) of China under Grant 60904006.
−0.4 −0.6 −0.8 0
5
10
15
20
Fig. 10. x1 , x2 .
0.4 0.2 0 −0.2 0.06 −0.4
0.04 0.02
−0.6
0 −0.02
−0.8 −1
2
4
0
6
8
5
10
12
14
10
16
18
15
20 20
Fig. 11. The e1 at different disturbance attenuation levels.
1.2 1
0.15
0.8
0.05
0.1
0
0.6
−0.05 2
0.4
4
6
8
10
12
14
12
14
16
18
20
0.2 0 −0.2 −0.4 −0.6
0
2
4
6
8
10
16
18
20
Fig. 12. The e2 at different disturbance attenuation levels.
saturations is considered and an adaptive radial basis neural network based controller which is designed with a priori consideration of actuator saturation effects and guarantees H1 tracking performance is proposed. Adaptive radial basis function neural networks are used to approximate the unknown nonlinearities. An auxiliary system is constructed to compensate the effects of actuator amplitude and rate saturations. A supervisory control is designed to attenuate the effects of extern disturbance and neural network approximation errors, so that the closed loop system achieves a prescribed H1 tracking performance. Analysis
References [1] Jing Zhou, Changyun Wen, Adaptive backstepping control of uncertain systems, in: Lecture Notes in Control and Information Sciences, Springer Berlin, vol. 372, 2008, pp. 189–197. [2] Marious, Jay Farrell, Manu Sharma, On-line approximation control of uncertain nonlinear systems: issues with control input saturation, in: Proceedings of the 2003 American Control Conference, Colorado, 2003, pp. 543–548. [3] Jay Farrell, Marios Polycarpou, Manu Sharma, Adaptive backstepping with magnitude, rate, and bandwidth constraints: aircraft longitude control, in: Proceedings of the 2003 American Control Conference, Colorado, 2003, pp. 3898–3904. [4] Jay Farrell, Marios Polycarpou, Manu Sharma, On-line approximation based control of uncertain nonlinear systems with magnitude, rate and bandwidth constraints on the states and actuators, in: Proceedings of the 2004 American Control Conference, Massachusetts, 2004, pp. 2557–2562. [5] Jay Farrell, Manu Sharma, Marious Polycarpou, Backstepping based flight control with adaptive function approximation, J. Guid. Control Dyn. 28 (6) (2005) 1089–1102. [6] Jay A. Farrell, Marios Polycarpou, Manu Sharma, Wenjie Dong, Command filtered backstepping, IEEE Trans. Automat. Control 54 (6) (2009) 1391–1395. [7] Wenjie Dong, Jay A. Farrell, Marios M. Polycarpou, Command filtered adaptive backstepping, IEEE Trans. Control Syst. Technol. 20 (3) (2012) 566–580. [8] L. Sonneveldt, Q.P. Chu, J.A. Mulder, Nonlinear flight control design using constrained adaptive backstepping, J. Guid. Control Dyn. 30 (2) (2007) 322–336. [9] L. Sonneveldt, Q.P. Chu, J.A. Mulder, Constrained adaptive backstepping flight control: application to a nonlinear F-16/MATV model, In: 2006 AIAA Guidance, Navigation, and Control Conference and Exhibit, Keystone, Colorado, USA, 2006. [10] Changyuan Wen, Jing Zhou, Zhitao Liu, Hongye Su, Robust adaptive control of uncertain nonlinear system in the presence of input saturation and external disturbance, IEEE Trans. Automat. Control 56 (7) (2011) 1672–1678. [11] A.H. Bajodah, Asymptotic generalised dynamic inversion attitude control, IET Control Theory Appl. 4 (5) (2010) 827–840. [12] S. Siberling, A.P. Chu, J.A. Mulder, Robust flight control using incremental nonlinear dynamic inversion and angular acceleration prediction, J. Guid. Control Dyn. 33 (6) (2010) 1732–1742. [13] Monish Deepak Tandale, Adaptive Dynamic Inversion Control of Nonlinear Systems Subjected to Control Saturation Constraints, Doctor dissertation, Texas A&M University, Texas, USA, 2007. [14] Keisuke Enomoto, A study on the dynamic inversion method with nonlinear function considering saturation, in: AIAA Guidance, Navigation, and Control Conference, Toronto, Ontario Canada, 2010. [15] Eugene Lavretsky, Naira Hovakimyan, Positive m-modification for stable adaptation in dynamic inversion based adaptive control with input saturation, in: Proceedings of the 2005 American Control Conference, Portland, OR, USA, 2005, pp. 3373–3378. [16] Guido Hermann, Matthew C. Turner, Prathyush Menon, et al., Anti-windup synthesis for nonlinear dynamic inversion controllers, in: Proceedings of the IFAC ROCOND, Toulouse, France, 2006. [17] Prathyush P Menon, Guido Herrmann, Matthew Turner, et al., Nonlinear dynamic inversion based anti-windup: an aerospace application, in: Proceedings of the 17-th Word Congress of The International Federation of Automatic Control, Seoul, Korea, 2008, pp. 14156–14161. [18] Q. Hu, G.P. Rngaiah, Anti-windup schemes for uncertain nonlinear systems, IEE Proc. Control Theory Appl. 147 (3) (2000) 321–329. [19] G. Cybenko, Approximation by superpositions of a sigmoidal function, Math. Control Signals Syst. 2 (4) (1989) 303–314. [20] Simon Haykin, Neural Networks: A Comprehensive Foundation, 2nd Edition, Prentice-Hall Inc., 1999. [21] Eric N. Johnson, Anthony J. Calise, Hesham A. El-Shirbiny, Feedback linearization with neural network augmentation applied to X-33 attitude control, AIAA-2000-4157, 2000. [22] Chen mou, Zou Jie, Feng Xing, Jiang Changsheng, Approximation based tracking control of uncertain MIMO nonlinear systems with input saturation, in: Proceedings of the 29th Chinese Control Conference, China, 2010, pp. 6155–6160.
80
R. Yuan et al. / Neurocomputing 125 (2014) 72–80
[23] Yuto Yuasa, Noboru Sakamoto, Yoshio Umemura, Optimal control design for systems with input saturations and rate limiters, in: 2010 SICE Annual Conference, 2010, pp. 2042–2045. [24] Murad Abu-Khalaf, Frank L. Lewis, Nearly optimal control laws for nonlinear systems with saturating actuators using a neural network HJB approach, Automatica 41 (5) (2005) 779–791. ¨ ¨ [25] Lars Grune, Jurgen Pannek, Nonlinear Model Predictive Control: Theory and Algorithms, Springer, 2011. [26] Bing Xiao, Qinglei Hu, Adaptive sliding mode fault tolerant attitude tracking control for flexible spacecraft under actuator saturation, IEEE Trans. Control Syst. Technol. (99) (2011) 1–8. [27] Ruyi Yuan, Jianqiang Yi, Wensheng Yu, Guoliang Fan, Adaptive controller design for uncertain nonlinear systems with input magnitude and rate limitations, in: Proceedings of the 2011 American Control Conference, USA, 2011, pp. 3536–3541. [28] R. Penrose, A generalized inverse for matrices, Proc. Cambridge Philos. Soc. 51 (3) (1955) 406–413. [29] P. Ioannou, J. Sun, Robust Adaptive Control, Prentice-Hall Inc., 1996.
Ruyi Yuan is a research assistant in the Integrated Information System Research Center, Institute of Automation, Chinese Academy of Sciences. He received B.S. degree from Hunan University in 2006, Ph.D. from Institute of Automation, Chinese Academy of Sciences in 2011. His current research interest is in the area of nonlinear control, adaptive control and flight control.
Xiangmin Tan is a research assistant in the Integrated Information System Research Center, Institute of Automation Chinese Academy of Sciences, China. His research interests lie in the area of robotics, industrial control, fuzzy systems, and neural networks.
Guoliang Fan is a Senior Engineer at the Institute of Automation, Chinese Academy of Sciences. His research interest covers non-minimum phase system, nonlinear control, robust adaptive control, and flight control.
Jianqiang Yi received the B.E. degree from the Beijing Institute of Technology, Beijing, China, in 1985, and the M.E. and Ph.D. degrees from the Kyushu Institute of Technology, Kitakyushu, Japan, in 1989 and 1992 respectively. From 1992 to 1994, he was with the Computer Software Development Company, Tokyo, Japan. From 1994 to 2001, he was a Chief Researcher at MYCOM Inc., Kyoto, Japan. Currently, he is a Full Professor in the Institute of Automation, Chinese Academy of Sciences. His research interests include theories and applications of intelligent control, intelligent robotics, underactuated system control, slidingmode control, flight control, etc. He is an Associate Editor for the IEEE Computational Intelligence Magazine, Journal of Advanced Computational Intelligence and Intelligent Informatics, and Journal of Innovative Computing, Information and Control. He worked as a Research Fellow at CSD, Inc., Tokyo, Japan from 1992 to 1994, and a Chief Engineer at MYCOM, Inc., Kyoto, Japan from 1994 to 2001. Since 2001 he has been with the Institute of Automation, Chinese Academy of Sciences, China, where he is currently a Professor. His research interests lie in the area of intelligent control and robotics.
